Israel Journal of Mathematics

, Volume 207, Issue 2, pp 617–651 | Cite as

Structural connections between a forcing class and its modal logic

  • Joel David Hamkins
  • George Leibman
  • Benedikt Löwe


Every definable forcing class Γ gives rise to a corresponding forcing modality \({\square _\Gamma }\) where \({\square _{\Gamma \varphi }}\) means that ϕ is true in all Γ extensions, and the valid principles of Γ forcing are the modal assertions that are valid for this forcing interpretation. For example, [10] shows that if ZFC is consistent, then the ZFC-provably valid principles of the class of all forcing are precisely the assertions of the modal theory S4.2. In this article, we prove similarly that the provably valid principles of collapse forcing, Cohen forcing and other classes are in each case exactly S4.3; the provably valid principles of c.c.c. forcing, proper forcing, and others are each contained within S4.3 and do not contain S4.2; the provably valid principles of countably closed forcing, CH-preserving forcing and others are each exactly S4.2; and the provably valid principles of ω1-preserving forcing are contained within S4.tBA. All these results arise from general structural connections we have identified between a forcing class and the modal logic of forcing to which it gives rise.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    U. Abraham, On forcing without the continuum hypothesis, Journal of Symbolic Logic 48 (1983), 658–661.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    P. Blackburn, M. de Rijke and Y. Venema, Modal Logic, Cambridge Tracts in Theoretical Computer Science, Vol. 53, Cambridge University Press, Cambridge, 2001.MATHGoogle Scholar
  3. [3]
    A. Chagrov and M. Zakharyaschev, Modal logic, Oxford Logic Guides, Vol. 35, Oxford University Press, New York, 1997.MATHGoogle Scholar
  4. [4]
    L. Esakia and B. Löwe, Fatal Heyting algebras and forcing persistent sentences, Studia Logica 100 (2012), 163–173.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    S. Friedman, S. Fuchino and H. Sakai, On the set-generic multiverse, submitted, 2012.Google Scholar
  6. [6]
    G. Fuchs, Closed maximality principles: implications, separations and combinations, Journal of Symbolic Logic 73 (2008), 276–308.MATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    G. Fuchs, Combined maximality principles up to large cardinals, Journal of Symbolic Logic 74 (2009), 1015–1046.MATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    D. M. Gabbay, The decidability of the Kreisel-Putnam system, Journal of Symbolic Logic 35 (1970), 431–437.MathSciNetCrossRefGoogle Scholar
  9. [9]
    J. D. Hamkins, A simple maximality principle, Journal of Symbolic Logic 68 (2003), 527–550.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    J. D. Hamkins and B. Löwe, The modal logic of forcing, Transactions of the American Mathematical Society 360 (2008), 1793–1817.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    J. D. Hamkins and B. Löwe, Moving up and down in the generic multiverse, in Logic and its Applications, Lecture Notes in Computer Science, Vol. 7750, Springer-Verlag, Heidelberg, 2013, pp. 139–147.CrossRefGoogle Scholar
  12. [12]
    J. D. Hamkins and W. H. Woodin, The necessary maximality principle for c.c.c. forcing is equiconsistent with a weakly compact cardinal, Mathematical Logic Quarterly 51 (2005), 493–498.MATHMathSciNetCrossRefGoogle Scholar
  13. [13]
    G. E. Hughes and M. J. Cresswell, A new Introduction to Modal Logic, Routledge, London, 1996.MATHCrossRefGoogle Scholar
  14. [14]
    T. C. Inamdar, On the modal logics of some set-theoretic constructions, Master’s thesis, Universiteit van Amsterdam, 2013, ILLC Publications MoL-2013-07.Google Scholar
  15. [15]
    T. Jech, Set Theory, 3rd ed., Springer Monographs in Mathematics, Springer-Verlag, Heidelberg, 2003.MATHGoogle Scholar
  16. [16]
    G. Leibman, Consistency strengths of modified maximality principles, Ph.D. thesis, City University of New York, 2004.Google Scholar
  17. [17]
    G. Leibman, The consistency strength of MPCCC(), Notre Dame Journal of Formal Logic 51 (2010), 181–193.MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    L. L. Maksimova, D. P. Skvorcov and Valentin B. Šehtman, Impossibility of finite axiomatization of Medvedev’s logic of finite problems, Doklady Akademii Nauk SSSR 245 (1979), 1051–1054.MathSciNetGoogle Scholar
  19. [19]
    C. J. Rittberg, The modal logic of forcing, Master’s thesis, Westfälische Wilhelms-Universität Münster, 2010.Google Scholar
  20. [20]
    J. Stavi and J. Väänänen, Reflection principles for the continuum, in Logic and Algebra, Contemporary Mathematics, Vol. 302, American Mathematical Society, Providence, RI, 2002, pp. 59–84.CrossRefGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2015

Authors and Affiliations

  • Joel David Hamkins
    • 1
    • 2
  • George Leibman
    • 3
  • Benedikt Löwe
    • 4
    • 5
  1. 1.Mathematics ProgramThe Graduate Center of The City University of New YorkNew YorkUSA
  2. 2.Department of MathematicsThe College of Staten Island of CUNYStaten IslandUSA
  3. 3.Department of Mathematics and Computer Science, Bronx Community CollegeThe City University of New YorkBronxUSA
  4. 4.Institute for Logic, Language and ComputationUniversiteit van AmsterdamAmsterdamThe Netherlands
  5. 5.Fachbereich MathematikUniversität HamburgHamburgGermany

Personalised recommendations